It is desirable in the bottling industry that the filling of containers with liquid be done efficiently and with reduced mess. The prior art includes attempts at accomplishing this. For example, expired U.S. Pat. No. 4,349,053 (the “'053 Eisenberg Patent”), the contents of which are hereby incorporated by reference in their entirety, discloses a method and apparatus for filling containers with liquid. Referring to the '053 Eisenberg Patent, and, as reproduced in part therefrom as FIGS. 1A and 1B of the present application, the '053 Eisenberg Patent discloses containers 10 set on pivoting mounts 12 with liquid flow being channeled from weirs 14 into the open tops of the containers 10. To reduce system complexity, the '053 Eisenberg Patent allows for an overflow condition, and excess liquid received by a container 10 is poured out into a trough 16 for recirculation ultimately back to weir 14. To reduce mess, e.g., to minimize contact between the liquid and an exterior of the containers 10 during an overfill, the containers 10 are tilted away from the liquid flow, though such provides a headspace 18 at the top of the container 10 that might be undesirably large. What is needed in the art are systems and methods for filling containers with a liquid so as to provide reduced headspace and reduced mess in connection with an overfill condition.